I have never heard of a bank that has a hard requirement of a mobile app. Certainly none of the major banks like Wells Fargo or Chase require one. I do not own a phone and managers at times have to come up with undocumented fallback methods, but there is always a way.
I cannot imagine a legal defense for forcing someone to accept the terms of service of Apple or Google to use their bank account.
I cannot imagine a legal defense for forcing someone to accept the terms of service of Apple or Google to use their bank account.